Refer to the equation A x^2+C y^2+D x+E y+F=0, where A and C...

refer to the equation $A x^2+C y^2+D x+E y+F=0$, where $A$ and $C$ are not both 0 .
Show that if $A=0, C \neq 0$, and $D \neq 0$, then the equation is equivalent to one of the form

$$
(y-k)^2=4 p(x-h)
$$

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They are represented by a general quadratic equation in two variables, and depending on the coefficients, the curve can be a parabola, ellipse, or hyperbola. Understanding the general form helps in identifying the type of conic and the necessary steps to convert it to its standard form.

Parabola

A parabola is a type of conic section characterized by having exactly one squared term in its standard form. The definition of a parabola involves the set of points equidistant from a fixed point (focus) and a fixed line (directrix). Recognizing the absence of one squared term in the general equation helps in identifying a parabola and understanding its geometric properties and orientation.

Completing the Square

Completing the square is a method used to convert quadratic expressions into a perfect square trinomial. This technique is essential in rewriting a conic section's equation into its standard form. It involves reorganizing and grouping terms related to a variable, factoring out coefficients if necessary, and adding and subtracting appropriate constants to obtain a complete square, thereby simplifying the analysis and graphing of the conic.

Vertex Form and Translation

The vertex form of a parabola, such as (y-k)² = 4p(x-h), clearly indicates the vertex (h, k) of the parabola and the focal distance p. This form results from a coordinate translation that repositions the parabola's vertex at the origin, making its geometric features more apparent. Understanding this transformation is key to linking the algebraic manipulation of the general equation to its geometric interpretation.

Please give Ace some feedback